Re: velocità nella dimensione tempo

From: Giorgio Bibbiani <giorgiobibbiani_at_TIN.it>
Date: Sun, 23 Jan 2022 07:26:14 +0100

Il 22/01/2022 17:31, Angelo M. ha scritto:
...
> Brian Green, ne: L'Universo Elegante, sostiene che si possa parlare di velocità di un oggetto nel tempo.

Si può parlare di tutto, ma alla fine ciò che conta sono le leggi quantitative, cioè per poter
parlare di "velocità di un oggetto nel tempo" occorre prima aver definito _operativamente_
cosa essa sia (è solo un altro nome per la derivata del tempo proprio rispetto a quello coordinato,
dtau/dt).

> Inoltre attribuisce addirittura ad Einstein l'affermazione "tutti gli oggetti dell'universo sono SEMPRE in moto nello spaziotempo con
> velocità fissa: quella della luce".

Sembra una attribuzione indiretta, non una citazione, v. il testo originale:

Since this view proclaims that space and time are simply different examples of dimensions, can we speak of an object's speed
through time in a manner resembling the concept of its speed through space? We can.
A big clue for how to do this comes from a central piece of information we have already encountered. When an object moves
through space relative to us, its clock runs slow compared to ours. That is, the speed of its motion through time slows down. Here's the leap:
Einstein proclaimed that all objects in the universe are always traveling through spacetime at one fixed speedâ€"that of light. This is a strange
idea; we are used to the notion that objects travel at speeds considerably less than that of light. We have repeatedly emphasized this as the
reason relativistic effects are so unfamiliar in the everyday world. All of this is true. We are presently talking about an object's combined
speed through all four dimensionsâ€"three space and one timeâ€"and it is the object's speed in this generalized sense that is equal to that of
light. To understand this more fully and to reveal its importance, we note that like the impractical single-speed car discussed above, this one
fixed speed can be shared between the different dimensionsâ€" different space and time dimensions, that is. If an object is sitting still
(relative to us) and consequently does not move through space at all, then in analogy to the first runs of the car, all of the object's motion
is used to travel through one dimensionâ€"in this case, the time dimension. Moreover, all objects that are at rest relative to us and to each
other move through timeâ€"they ageâ€"at exactly the same rate or speed. If an object does move through space, however, this means that some of the
previous motion through time must be diverted. Like the car traveling at an angle, this sharing of motion implies that the object will travel
more slowly through time than its stationary counterparts, since some of its motion is now being used to move through space. That is, its clock
will tick more slowly if it moves through space. This is exactly what we found earlier. We now see that time slows down when an object moves
relative to us because this diverts some of its motion through time into motion through space. The speed of an object through space is thus
merely a reflection of how much of its motion through time is diverted.

For the mathematically inclined reader, we note that from the spacetime position 4-vector x = (ct, x1, x2, x3) = (ct, xâ†') we can produce the
velocity 4-vector u = dx/dτ, where τ is the proper time defined by dτ^2 = dt^2 - c^-2(dx1^2 + dx2^2 + dx3^2). Then, the "speed through
spacetime" is the magnitude of the 4-vector u, √(((c^2dt^2 - dxâ†'^2) / (dt^2 - c^-2 dxâ†'^2))), which
is identically the speed of light, c. Now, we can rearrange the equation c^2(dt/dÏ„)^2 - (dxâ†'/dÏ„)^2 = c^2, to be
c^2(dÏ„/dt)^2 + (dxâ†'/dt)^2 = c^2. This shows that an increase in an object's speed through space, √((dxâ†'/dt)^2) must be accompanied by a decrease
in dτ/dt, the latter being the object's speed through time (the rate at which time elapses on its own clock, dτ, as compared with that on our
stationary clock, dt).

Ciao

-- 
Giorgio Bibbiani
Received on Sun Jan 23 2022 - 07:26:14 CET

This archive was generated by hypermail 2.3.0 : Thu Nov 21 2024 - 05:10:01 CET